Guardado en:
| Autores principales: | , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2025
|
| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2509.05600 |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
Tabla de Contenidos:
- Sharp Fourier restriction theory and finite field extension theory have both been topics of interest in the last decades. Very recently, in \cite{GonzalezOliveira}, the research into the intersection of these two topics started. There it was established that, for the $(3,1)$-cone $Γ_{(3,1)}^3:=\{\boldsymbolη\in \mathbb{F}_q^4\setminus\{\boldsymbol{0}\} : η_1^2+η_2^2+η_3^2=η_4^2\},$ the Fourier extension map from $L^2\to L^{4}$ is maximized by constant functions when $q=3\, \pmod{4}$. In this manuscript, we advance this line of inquiry by establishing sharp inequalities for the $L^{2}\to L^{4}$ extension inequalities applicable for all remaining cones $Γ^3\subset \mathbb{F}_q^4$. These cones include the $(2,2)$-cone $Γ_{(2,2)}^3:=\{\boldsymbolη\in \mathbb{F}_q^4\setminus\{\boldsymbol{0}\} : η_1^2+η_2^2=η_3^2+η_4^2\}$ for general $q=p^n$ and the $(3,1)$-cone when $q=1\, \pmod{4}$. Moreover, we classify all the extremizers in each case. We note that the analogous problem for the $(2, 2)$-cone in the euclidean setting remains open.